Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem 16: Calculus Exercise**
*Objective:*
1. Find the maximum and minimum values for the function \( f(x) \).
2. Estimate the integral \( \int_{1}^{3} \sqrt{x^2 + 1} \, dx \).
*Instructions:*
- Start by determining the critical points of \( f(x) \) to find its maximum and minimum values. Use derivative tests to confirm if these points are indeed maxima or minima.
- Once you have found these values, proceed to estimate the given definite integral.
- Consider the behavior of \( \sqrt{x^2 + 1} \) over the interval [1, 3] and apply appropriate numerical methods for estimation if necessary (e.g., trapezoidal rule, Simpson's rule).
The task involves both optimization and integration techniques commonly encountered in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa17b68c5-364b-49f9-8889-84e65ef1bef7%2F1a65bf02-2aab-469f-ac97-a8f99f3ebc75%2F8q8xf9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 16: Calculus Exercise**
*Objective:*
1. Find the maximum and minimum values for the function \( f(x) \).
2. Estimate the integral \( \int_{1}^{3} \sqrt{x^2 + 1} \, dx \).
*Instructions:*
- Start by determining the critical points of \( f(x) \) to find its maximum and minimum values. Use derivative tests to confirm if these points are indeed maxima or minima.
- Once you have found these values, proceed to estimate the given definite integral.
- Consider the behavior of \( \sqrt{x^2 + 1} \) over the interval [1, 3] and apply appropriate numerical methods for estimation if necessary (e.g., trapezoidal rule, Simpson's rule).
The task involves both optimization and integration techniques commonly encountered in calculus.
Expert Solution
Step 1
We know that trigonometry property :
Sec2x = 1+tan2x .
We know that general:
1.integration of secx is ln(secx+tanx)+C.
2.derivative of d(xn)/dx = nxn-1.
Here i attended the solution of this question.
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